When each side of a cube is increased by 2cm, the volume is increased by ` 1016 cm^(3) `. Find the side of the cube. If each side of it is decreased by 2 cm, by how much will the volume decrease ?

Options

A. 12 cm , `729 cm^(3) `
B. `8 cm, 512 cm^(3) `
C. `9 cm , 729 cm^(3) `
D. `12 cm, 728 cm^(3) `

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Understand the problem
We know that when each side of a cube is increased by 2 cm, the volume increases by 1016 cm³. We need to find the original side length of the cube.

### Step 2: Set up the equation
Let the original side length of the cube be \( x \) cm. The volume of a cube is given by the formula:
\[ \text{Volume} = \text{side}^3 = x^3 \]
When the side is increased by 2 cm, the new side length becomes \( x + 2 \) cm. The new volume is:
\[ \text{New Volume} = (x + 2)^3 \]
According to the problem, the increase in volume is 1016 cm³, so we can set up the equation:
\[ (x + 2)^3 - x^3 = 1016 \]

### Step 3: Expand the equation
Now, we will expand \( (x + 2)^3 \):
\[ (x + 2)^3 = x^3 + 3(x^2)(2) + 3(x)(2^2) + 2^3 = x^3 + 6x^2 + 12x + 8 \]
Substituting this back into the equation gives:
\[ x^3 + 6x^2 + 12x + 8 - x^3 = 1016 \]
This simplifies to:
\[ 6x^2 + 12x + 8 = 1016 \]

### Step 4: Rearrange the equation
Now, we will rearrange the equation:
\[ 6x^2 + 12x + 8 - 1016 = 0 \]
This simplifies to:
\[ 6x^2 + 12x - 1008 = 0 \]

### Step 5: Divide the equation by 6
To simplify, we divide the entire equation by 6:
\[ x^2 + 2x - 168 = 0 \]

### Step 6: Factor the quadratic equation
Next, we will factor the quadratic equation:
\[ x^2 + 14x - 12x - 168 = 0 \]
This can be factored as:
\[ (x + 14)(x - 12) = 0 \]

### Step 7: Solve for \( x \)
Setting each factor to zero gives us:
1. \( x + 14 = 0 \) → \( x = -14 \) (not valid since side length cannot be negative)
2. \( x - 12 = 0 \) → \( x = 12 \)

Thus, the side length of the cube is \( 12 \) cm.

### Step 8: Calculate the original volume
Now we calculate the original volume of the cube:
\[ \text{Volume} = x^3 = 12^3 = 1728 \, \text{cm}^3 \]

### Step 9: Calculate the new volume when the side is decreased by 2 cm
If each side is decreased by 2 cm, the new side length is:
\[ 12 - 2 = 10 \, \text{cm} \]
The new volume is:
\[ \text{New Volume} = 10^3 = 1000 \, \text{cm}^3 \]

### Step 10: Calculate the decrease in volume
The decrease in volume is:
\[ \text{Decrease} = 1728 - 1000 = 728 \, \text{cm}^3 \]

### Final Answer
The side of the cube is \( 12 \, \text{cm} \) and the decrease in volume when each side is decreased by 2 cm is \( 728 \, \text{cm}^3 \).

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